📚 IGCSE CAIE Geography: Formula & Theorem Quick Reference Handbook | IGCSE CAIE地理：公式定理速查手册
This handbook compiles essential formulas, calculations and key models you need to master for the IGCSE CAIE Geography examination. It covers map skills, river studies, population, settlement, climate and economic development. Use this page as your rapid revision companion.
本手册汇总了IGCSE CAIE地理考试必须掌握的核心公式、计算方法和关键模型，涵盖地图技能、河流、人口、聚落、气候和经济发展。将此页面作为你的快速复习伴侣。
1. Map Skills: Scale, Distance & Area | 地图技能：比例尺、距离与面积
Map scales can be given as a statement (1 cm to 1 km), a representative fraction (RF) 1:100,000 or a linear scale. Calculations of real-world distance and area are fundamental.
地图比例尺可以表示为文字式（1厘米代表1千米）、分数式1:100,000或直线比例尺。实地距离与面积的计算是基础技能。
The table below summarises the three core map-work formulas. Always ensure you use the same units before multiplying.
下表汇总了三个核心地图作业公式。务必在相乘之前统一单位。
| Formula (English) | 公式（中文） |
|---|---|
| Actual distance = Map distance × Scale denominator | 实地距离 = 图上距离 × 比例尺分母 |
| Area on the ground = Map area × (Scale denominator)² | 实地面积 = 图上面积 × (比例尺分母)² |
| Scale = Map distance ÷ Actual distance (in same units) | 比例尺 = 图上距离 ÷ 实地距离（相同单位） |
When measuring a curved distance on a map, use a piece of string along the feature, then straighten it against the linear scale.
测量地图上的曲线距离时，可用细绳沿地物摆放，再将细绳拉直，与直线比例尺比对。
2. Gradient and Cross-Section Profile | 坡度和剖面图
Gradient compares the vertical rise to the horizontal distance. It is often expressed as a ratio (1 : X).
坡度表示垂直抬升与水平距离的比值，通常写成分数比形式1 : X。
Gradient = Vertical Interval (VI) ÷ Horizontal Equivalent (HE)
The vertical interval is taken from contour lines (difference in height). The horizontal equivalent is the ground distance between those two points, converted to the same unit as the VI.
垂直间距取自等高线的高差，水平距离是两点间的实地距离，需换算为与垂直间距相同的单位。
To draw a cross‑section, place a strip of paper along the section line on the map, mark every contour crossing, project the heights onto graph paper and join the points smoothly.
绘制剖面图时，将纸条沿剖面线放在地图上，标记每条等高线交点，将高程投影到方格纸上，最后用平滑曲线连接各点。
3. River Discharge and Velocity | 河流流量与流速
River measurements are essential for flood risk and channel analysis. Discharge (Q) is the volume of water passing a point per second, measured in cubic metres per second (m³/s).
河流测量对洪水风险与河道分析至关重要。流量(Q)是每秒通过某断面的水量，单位为立方米每秒（m³/s）。
Discharge (Q) = Cross‑sectional area (A) × Average velocity (V)
Cross‑sectional area is calculated by multiplying channel width by mean depth. Velocity is obtained by timing a float over a known distance, then applying a correction factor (usually 0.85 for a surface float).
过水断面面积由河宽乘以平均水深求得。流速通过测量浮标漂过固定距离的时间来估算，再乘以修正系数（水面浮标通常取0.85）。
Always take several velocity readings across the channel and average them. Units must be consistent: width and depth in metres, time in seconds.
务必在河宽方向多次测量流速并取平均值。单位必须一致：宽度与深度用米，时间用秒。
4. Climate Calculations: Mean Temperature, Range and Total Rainfall | 气候计算：平均气温、气温较差和总降水量
Climate data is summarised using averages and totals. These simple statistics allow you to compare stations and construct climate graphs.
气候数据通过平均值和总量来概括。这些简单的统计量可用于比较不同测站并绘制气候图。
Mean annual temperature = (Sum of 12 monthly mean temperatures) ÷ 12
Annual temperature range = Highest monthly mean temperature − Lowest monthly mean temperature
Total annual precipitation = Sum of all monthly precipitation totals
Use a bar‑and‑line climate graph: temperature as a red line, rainfall as blue bars. Always label axes and give a title.
使用柱线混合气候图：气温用红色折线，降水量用蓝色柱状。务必标注坐标轴和图名。
For seasonal comparisons you can also calculate mean temperature of the three warmest months and the three coldest months.
进行季节对比时，还可计算最热三个月和最冷三个月的平均气温。
5. Population Density, Crude Birth Rate and Crude Death Rate | 人口密度、粗出生率和粗死亡率
These basic demographic indicators describe the pressure on land and the pace of natural replacement.
这些基本人口指标用于描述土地压力和自然更替的速度。
Population density = Total population ÷ Land area (km²)
Crude Birth Rate (CBR) = (Number of live births in one year ÷ Mid‑year total population) × 1000
Crude Death Rate (CDR) = (Number of deaths in one year ÷ Mid‑year total population) × 1000
Birth and death rates are expressed per 1000 people to allow comparisons between countries of different sizes. Remember ‘crude’ means they do not account for age structure.
出生率和死亡率以千分率表示，以便在不同规模的国家之间进行比较。注意“粗”是因其未考虑年龄结构。
6. Natural Population Change and Growth Rate | 自然人口变化与增长率
Natural change ignores migration and focuses on births and deaths. The overall growth rate includes net migration.
自然变化忽略迁移，只关注出生与死亡。总增长率则包含净迁移。
Natural increase rate (per 1000) = CBR − CDR
Population growth rate (%) = [(Births − Deaths + Net migration) ÷ Mid‑year population] × 100
A negative natural increase occurs when deaths exceed births. Doubling time can be estimated using the ‘rule of 70’: Doubling time (years) = 70 ÷ Growth rate (%).
当死亡人数超过出生人数时，出现负自然增长。可用“70法则”估算倍增时间：倍增时间（年）= 70 ÷ 增长率(%)。
7. Dependency Ratio | 依赖比
The dependency ratio compares the economically dependent population to the working‑age population. It gives an indication of the burden on the productive sector.
依赖比将经济依赖人口与劳动年龄人口进行比较，反映生产部门需要供养的负担。
Dependency ratio = [(Population 0–14 + Population 65 and over) ÷ Population 15–64] × 100
A high dependency ratio (e.g. >80) means fewer workers are supporting many young and elderly dependents. Many LICs have high youth dependency; some HICs face high old‑age dependency.
高依赖比（如超过80）表示较少劳动者需供养大量少儿和老人。许多低收入国家青年依赖比高，而一些高收入国家则面临高老年依赖比。
8. Urbanisation Rate and Level of Urbanisation | 城市化率与城市化水平
The level of urbanisation is the share of people living in urban areas, while the rate of urbanisation measures how fast this share is growing.
城市化水平指居住在城镇地区的人口比例，城市化速度则衡量这一比例的增长快慢。
Level of urbanisation = (Urban population ÷ Total population) × 100
Average annual urban growth rate = [(Urban pop year 2 − Urban pop year 1) ÷ Number of years] ÷ Urban pop year 1 × 100
Rapid urbanisation is common in emerging economies. Note that a city can have a high urban growth rate even if the national level of urbanisation is still low.
快速城市化常见于新兴经济体。需注意，即使全国城市化水平较低，某一城市的增速也可能很高。
9. Key Model: Demographic Transition Model (DTM) | 关键模型：人口转变模型
The Demographic Transition Model explains how birth rates, death rates and population size change as a country develops. Although not a formula, it is a fundamental ‘theorem’ of population geography.
人口转变模型解释了一个国家发展过程中出生率、死亡率和人口规模的演变。虽非公式，却是人口地理学的基本“定理”。
The five stages are summarised below.
五个阶段的特征总结如下。
| Stage | Birth rate | Death rate | Population change | Examples |
|---|---|---|---|---|
| 1 | High and fluctuating | High and fluctuating | Slow increase | Remote tribal groups |
| 2 | High | Falling rapidly | Very rapid increase | Afghanistan, Niger |
| 3 | Falling | Falling slowly | Increase slows | India, Bangladesh |
| 4 | Low | Low | Stable or slow growth | USA, France |
| 5 | Very low | Low (may rise slightly) | Decline | Japan, Germany |
Understanding the DTM helps you explain population pyramids, government policies and future development challenges.
理解人口转变模型有助于解释人口金字塔、政府政策和未来的发展挑战。
10. GDP per Capita and Crop Yield | 人均GDP与作物产量
Economic and agricultural indicators complement demographic data when comparing development and food security.
在进行发展和粮食安全对比时，经济与农业指标可与人口数据相互补充。
GDP per capita = Gross Domestic Product ÷ Total population
GDP per capita gives an average income and is often expressed in US dollars (US$). It does not show inequality within the country.
人均GDP反映平均收入，通常以美元表示，但无法体现国家内部的贫富差距。
Crop yield = Total production (kg or tonnes) ÷ Cultivated area (hectares)
High yields indicate intensive farming or favourable conditions, while low yields may point to subsistence agriculture or poor soil fertility.
高产量说明耕作集约或条件优越，低产量则可能指向自给农业或贫瘠的土壤肥力。
For livestock farming, you may also encounter stocking density: number of animals per hectare of grazing land.
在畜牧业中，还可能遇到载畜密度：每公顷草场承载的牲畜头数。
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